Probability theory plays role in all studies of natural processes across scientific disciplines. The need for a theoretical probabilistic foundation is obvious since natural variation effects all measurements, observations and findings about different phenomena. Probability theory provides the basic techniques for statistical inference.

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Probability theory plays a role in all studies of natural processes across scientific disciplines. The need for a theoretical probabilistic foundation is obvious since natural variation affects all measurements, observations and findings about different phenomena. Probability theory provides the basic techniques for statistical inference.

===Random Sampling===

===Random Sampling===

A '''simple random sample''' of ''n'' items is a sample in which every member of the population has an equal chance of being selected ''and'' the members of the sample are chosen independently.

A '''simple random sample''' of ''n'' items is a sample in which every member of the population has an equal chance of being selected ''and'' the members of the sample are chosen independently.

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* '''Example''': Consider a class of students as the population under study. If we select a sample of size 5, each possible sample of size 5 must have the same chance of being selected. When a sample is chosen randomly, it is the process of selection that is random. How could we select five members from this class randomly? Random sampling from finite (or countable) populations is well-defined. On the contrary, random sampling of uncountable populations is only allowed under the [http://en.wikipedia.org/wiki/Axiom_of_Choice Axiom of Choice].

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* '''Example''': Consider a class of students as the population under study. If we select a sample of size 5, each possible sample of size 5 must have the same chance of being selected. When a sample is chosen randomly, the process of selection that is random. How could we select five members from this class randomly? Random sampling from finite (or countable) populations is well-defined. On the contrary, random sampling of uncountable populations is only allowed under the [http://en.wikipedia.org/wiki/Axiom_of_Choice Axiom of Choice].

* '''Random Number Generation''' using [[SOCR]]: You can use [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] to [[SOCR_EduMaterials_Activities_RNG | construct random samples]] of any size from a large number of distribution families.

* '''Random Number Generation''' using [[SOCR]]: You can use [http://socr.ucla.edu/htmls/SOCR_Modeler.html SOCR Modeler] to [[SOCR_EduMaterials_Activities_RNG | construct random samples]] of any size from a large number of distribution families.

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* '''Questions''':

* '''Questions''':

**How would you go about randomly selecting five students from a class of 100?

**How would you go about randomly selecting five students from a class of 100?

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**How representative of the population is the sample likely to be? The sample won’t exactly resemble the population as there will be some chance variation. This discrepancy is called chance '''error due to sampling'''.

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**How likely the sample is to represent of the population? The sample won’t exactly resemble the population as there will be some chance of variation. This discrepancy is called chance '''error due to sampling'''.

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* '''Definition''': '''Sampling bias''' is non-randomness that refers to some members having a tendency to be selected more readily than others. When the sample is biased the statistics turn out to be poor estimates.

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* '''Definition''': '''Sampling bias''' is non-random and refers to some members having a tendency to be selected more readily than others. When the sample is biased the statistics turn out to be poor estimates.

===Hands-on activities===

===Hands-on activities===

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** '''Swap''' strategy - choose one card first, as the computer reveals one of the donkey cards, you always swap your original guess and go with the third face-down card!

** '''Swap''' strategy - choose one card first, as the computer reveals one of the donkey cards, you always swap your original guess and go with the third face-down card!

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* You can try the [http://socr.ucla.edu/htmls/SOCR_Experiments.html Monty Hall Experiment], as well. There you can run a very large number of trials automatically and empirically observe the outcomes. Notice that your chance to win doubles if you use the swap-strategy. Why is that?

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* You can try the [http://socr.ucla.edu/htmls/SOCR_Experiments.html Monty Hall Experiment] as well. There you can run a very large number of trials automatically and observe the outcomes empirically. Notice that your chance to win doubles if you use the swap-strategy. Why is that?

* See the [[AP_Statistics_Curriculum_2007_Prob_Rules#Monty_Hall_Problem | conditional probability derivation]] of the exact chance of success.

===Law of Large Numbers===

===Law of Large Numbers===

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When studying the behavior of coin tosses, the law of large numbers implies that the relative proportion (relative frequency) of heads-to-tails in a coin toss experiment becomes more and more stable as the number of tosses increases. This regards the relative frequencies, not absolute counts of heads and tails.

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When studying the behavior of coin tosses, the law of large numbers implies that the relative proportion (relative frequency) of heads-to-tails in a coin toss experiment becomes more and more stable as the number of tosses increases. This regards to the relative frequencies, not absolute counts of heads and tails.

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* There are two widely held ''misconceptions'' about what the law of large numbers about coin tosses:

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* There are two widely held ''misconceptions'' about the law of large numbers relating to coin tosses:

**Differences between the actual numbers of heads and tails become more variable with increase of the number of tosses – a sequence of 10 heads doesn’t increase the chances of selecting a tail on the next trial.

**Differences between the actual numbers of heads and tails become more variable with increase of the number of tosses – a sequence of 10 heads doesn’t increase the chances of selecting a tail on the next trial.

** Coin toss results are independent and fair, and the outcome behavior is unpredictable.

** Coin toss results are independent and fair, and the outcome behavior is unpredictable.

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* '''First axiom''': The probability of an event is a non-negative real number: <math>P(E)\geq 0</math>, <math>\forall E\subseteq S</math>, where <math>S</math> is the sample space.

* '''First axiom''': The probability of an event is a non-negative real number: <math>P(E)\geq 0</math>, <math>\forall E\subseteq S</math>, where <math>S</math> is the sample space.

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* '''Second axiom''': This is the assumption of '''unit measure''': that the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space: <math>P(S) = 1</math>. This is often overlooked in some mistaken probability calculations; if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.

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* '''Second axiom''': This is the assumption of '''unit measure''': the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space: <math>P(S) = 1</math>. This is often overlooked in some mistaken probability calculations if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.

This is an ''approximate solution'' to the Birthday problem, you can also see the ''exact solution'' in the [[SOCR_EduMaterials_Activities_BirthdayExperiment#Exact_Calculations | Birthday Paradox Activity]].

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This is an ''approximate solution'' to the Birthday problem. You can also see the ''exact solution'' in the [[SOCR_EduMaterials_Activities_BirthdayExperiment#Exact_Calculations | Birthday Paradox Activity]].

===Examples===

===Examples===

====Elementary Probability====

====Elementary Probability====

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Each of the three boxes below contains two types of balls (''Red'' and ''Green''). Box 1 has ''4 Red'' and ''3 Green'' balls, box 2 has ''3 Red'' and ''2 Green'' balls, and box 3 has ''2 Red'' and ''1 Green'' balls. All balls are identical except for their labels. Which of the three boxes are you most likely to draw a ''Red'' ball from? In other words, if a randomly drawn ball is known to be Red, which box is the one that we most likely drew the ball out of?

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Each of the three boxes below contains two types of balls (''Red'' and ''Green''). Box 1 has ''4 Red'' and ''3 Green'' balls, box 2 has ''3 Red'' and ''2 Green'' balls, and box 3 has ''2 Red'' and ''1 Green'' balls. All balls are identical except for their labels. Which of the three boxes do you most likely to draw a ''Red'' ball from? In other words, if a randomly drawn ball is known to be Red, which box is the one that we most likely draw the ball out of?

* ''Exact solution'': The probabilities of drawing a Red ball out of each of the 3 boxes are 4/7, 3/5 and 2/3, respectively. The smallest common denominator of the prime numbers 7, 5 and 3 is their product 105. Thus, these probabilities may also be expressed as 60/105, 63/105 and 70/105, respectively. Clearly, the highest chance of drawing a Red ball is associated with box 3, despite the fact that this box has the smallest number of Red balls.

* ''Exact solution'': The probabilities of drawing a Red ball out of each of the 3 boxes are 4/7, 3/5 and 2/3, respectively. The smallest common denominator of the prime numbers 7, 5 and 3 is their product 105. Thus, these probabilities may also be expressed as 60/105, 63/105 and 70/105, respectively. Clearly, the highest chance of drawing a Red ball is associated with box 3, despite the fact that this box has the smallest number of Red balls.

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* ''Empirical solution'': This problem may also be explored experimentally using the [http://socr.ucla.edu/htmls/exp/Ball_and_Urn_Experiment.html SOCR Ball-and-Urn Experiment] (see [[SOCR_EduMaterials_Activities_BallAndRunExperiment |this activity]]). Each of the 3 figures below illustrate this experiment, where we sample (''with replacement'') 100 balls from each box, respectively.

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* ''Empirical solution'': This problem may also be explored experimentally using the [http://socr.ucla.edu/htmls/exp/Ball_and_Urn_Experiment.html SOCR Ball-and-Urn Experiment] (see [[SOCR_EduMaterials_Activities_BallAndRunExperiment |this activity]]). Each of the 3 figures below illustrates this experiment, where we sample (''with replacement'') 100 balls from each box, respectively.

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** Box 1: To empirically test the chance of drawing a Red ball from box 1, we set N=7 (total number of balls), R=4 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). The result of these 100 random draws will vary each time, however one such experiment generated 62 Red balls out of the 100 draws (see image below and the value of the ''Y=62'' variable in the summary table). <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig1.jpg|400px]]</center>

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** Box 1: To empirically test the chance of drawing a Red ball from box 1, we set N=7 (total number of balls), R=4 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). The result of these 100 random draws will vary each time. However one such experiment generated 62 Red balls out of the 100 draws (see image below and the value of the ''Y=62'' variable in the summary table). <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig1.jpg|400px]]</center>

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** Box 2: Now we set N=5 (total number of balls), R=3 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time, however one such experiment generated 64 Red balls out of the 100 draws (see image below and the value of the variable ''Y=64'' in the summary table below the graph). <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig2.jpg|400px]]</center>

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** Box 2: Now we set N=5 (total number of balls), R=3 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time. However one such experiment generated 64 Red balls out of the 100 draws (see image below and the value of the variable ''Y=64'' in the summary table below the graph). <center>[[Image:SOCR_EBook_Dinov_Prob_BallUrn_040308_Fig2.jpg|400px]]</center>

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** Box 3: Finally, we can set N=3 (total number of balls), R=2 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time, however one such experiment generated 71 Red balls out of the 100 draws (see image below and the value of the variable ''Y=71'' in the summary table below the graph).

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** Box 3: Finally, we can set N=3 (total number of balls), R=2 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time. However one such experiment generated 71 Red balls out of the 100 draws (see image below and the value of the variable ''Y=71'' in the summary table below the graph).

Fundamentals of Probability Theory

Probability theory plays a role in all studies of natural processes across scientific disciplines. The need for a theoretical probabilistic foundation is obvious since natural variation affects all measurements, observations and findings about different phenomena. Probability theory provides the basic techniques for statistical inference.

Random Sampling

A simple random sample of n items is a sample in which every member of the population has an equal chance of being selected and the members of the sample are chosen independently.

Example: Consider a class of students as the population under study. If we select a sample of size 5, each possible sample of size 5 must have the same chance of being selected. When a sample is chosen randomly, the process of selection that is random. How could we select five members from this class randomly? Random sampling from finite (or countable) populations is well-defined. On the contrary, random sampling of uncountable populations is only allowed under the Axiom of Choice.

How would you go about randomly selecting five students from a class of 100?

How likely the sample is to represent of the population? The sample won’t exactly resemble the population as there will be some chance of variation. This discrepancy is called chance error due to sampling.

Definition: Sampling bias is non-random and refers to some members having a tendency to be selected more readily than others. When the sample is biased the statistics turn out to be poor estimates.

Hands-on activities

Monty Hall (Three-Door) Problem: Go to SOCR Games and select the Monty Hall Game. Click in the Information button to get the instructions on using the applet. Run the game 10 times with one of two strategies:

Stay-home strategy - choose one card first, as the computer reveals one of the donkey cards, you always stay with the card you originally chose.

Swap strategy - choose one card first, as the computer reveals one of the donkey cards, you always swap your original guess and go with the third face-down card!

You can try the Monty Hall Experiment as well. There you can run a very large number of trials automatically and observe the outcomes empirically. Notice that your chance to win doubles if you use the swap-strategy. Why is that?

Law of Large Numbers

When studying the behavior of coin tosses, the law of large numbers implies that the relative proportion (relative frequency) of heads-to-tails in a coin toss experiment becomes more and more stable as the number of tosses increases. This regards to the relative frequencies, not absolute counts of heads and tails.

There are two widely held misconceptions about the law of large numbers relating to coin tosses:

Differences between the actual numbers of heads and tails become more variable with increase of the number of tosses – a sequence of 10 heads doesn’t increase the chances of selecting a tail on the next trial.

Coin toss results are independent and fair, and the outcome behavior is unpredictable.

Event Manipulations

Just like we develop rules for numeric arithmetic, we like to use certain event-manipulation rules (event-arithmetic).

Complement: The complement of an event A, denoted Ac or A', occurs if and only if A does not occur.

, read "A or B", contains all outcomes in A or B (or both).

, read "A and B", contains all outcomes which are in both A and B.

Draw Venn diagram pictures of these composite events.

Mutually exclusive events cannot occur at the same time ().

Axioms of probability

First axiom: The probability of an event is a non-negative real number: , , where S is the sample space.

Second axiom: This is the assumption of unit measure: the probability that some elementary event in the entire sample space will occur is 1. More specifically, there are no elementary events outside the sample space: P(S) = 1. This is often overlooked in some mistaken probability calculations if you cannot precisely define the whole sample space, then the probability of any subset cannot be defined either.

Third axiom: This is the assumption of additivity: Any countable sequence of pair-wise disjoint events E1,E2,... satisfies

Note: For a finite sample space, a sequence of number {} is a probability distribution for a sample space S = {}, if the probability of the outcome sk, p(sk) = pk, for each , all and .

Birthday Paradox

In a random group of N people, what is the probability, P, that at least two people have the same birthday?

Example, if N=23, P>0.5. Main confusion arises from the fact that in real life we rarely meet people having the same birthday as us, and we meet more than 23 people.

The reason for such high probability is that any of the 23 people can compare their birthday with any other one, not just you comparing your birthday to anybody else’s.

There are N-Choose-2 = 20*19/2 ways to select a pair of people from a group of 20. Assume there are 365 days in a year, P(one-particular-pair-same-B-day)=1/365, and
P(one-particular-pair-failure)=1-1/365 ~ 0.99726.

Examples

Elementary Probability

Each of the three boxes below contains two types of balls (Red and Green). Box 1 has 4 Red and 3 Green balls, box 2 has 3 Red and 2 Green balls, and box 3 has 2 Red and 1 Green balls. All balls are identical except for their labels. Which of the three boxes do you most likely to draw a Red ball from? In other words, if a randomly drawn ball is known to be Red, which box is the one that we most likely draw the ball out of?

Exact solution: The probabilities of drawing a Red ball out of each of the 3 boxes are 4/7, 3/5 and 2/3, respectively. The smallest common denominator of the prime numbers 7, 5 and 3 is their product 105. Thus, these probabilities may also be expressed as 60/105, 63/105 and 70/105, respectively. Clearly, the highest chance of drawing a Red ball is associated with box 3, despite the fact that this box has the smallest number of Red balls.

Empirical solution: This problem may also be explored experimentally using the SOCR Ball-and-Urn Experiment (see this activity). Each of the 3 figures below illustrates this experiment, where we sample (with replacement) 100 balls from each box, respectively.

Box 1: To empirically test the chance of drawing a Red ball from box 1, we set N=7 (total number of balls), R=4 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). The result of these 100 random draws will vary each time. However one such experiment generated 62 Red balls out of the 100 draws (see image below and the value of the Y=62 variable in the summary table).

Box 2: Now we set N=5 (total number of balls), R=3 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time. However one such experiment generated 64 Red balls out of the 100 draws (see image below and the value of the variable Y=64 in the summary table below the graph).

Box 3: Finally, we can set N=3 (total number of balls), R=2 (number of red balls in the population, and n=100 (number of balls we sample, i.e., number of experiments we do). Again, the result of these 100 random draws will vary each time. However one such experiment generated 71 Red balls out of the 100 draws (see image below and the value of the variable Y=71 in the summary table below the graph).

From these empirical tests, we can propose that a Red ball is most likely to be drawn from Box 3. Again, remember that these simulations will produce different results each time we do the experiments. However, the expected means (expected number of Red balls in sample of 100), which are reported in the bottom-right tables (for each box setting), indicate this conclusion more reliably. These expected Red ball counts for the 3 boxes are 57.14, 60.0 and 66.67, respectively.

SOCR Empirical Probabilities

For each of these hands-on interactive experiments discuss the sample space, probabilities, events of interest, event operations, and how to compute theoretically and empirically the probabilities of these events.